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Gabarito da lista 1 
Prof. Fernando Cesar Coelho França 
1) 
a) [1,5] 
b) Dom(r(t))={𝑡 ∈ ℝ/𝑡 > 0 𝑒 𝑡 ≠ 1} 
2) 
a) (1,0,0) 
b) (1, 1/2, 3) 
c) 2i+(1/2)j+tg 1 k 
3) 
a)v=(1,2) a=(0,2) 
b)v=(4,-4) a=(-16,-16) 
c)v=(3,4) a=(3,8) 
d)v=(0,6) a=(-4,0) 
4 
a)r’(t)=(
1
2√𝑡−1
, −
1
2√2−𝑡
), r’’(t)=(−
1
4(𝑡−1)
3
2
, −
1
4(2−𝑡)
3
2
) 
b)r’(t)=(
−1
𝑡2
, 3𝑐𝑜𝑠3𝑡), r’’(t)=(
2
𝑡3
, −9𝑠𝑒𝑛3𝑡) 
c)r’(t)=(
1
2√𝑡
, 2𝑒2𝑡, 1), r’’(t)=(−
1
4(𝑡)
3
2
, 4𝑒2𝑡, 0) 
d)r’(t)=(−
1
1−𝑡
, 𝑐𝑜𝑠𝑡, 2𝑡), r’’(t)=(−
1
(1−𝑡)2
, −𝑠𝑒𝑛𝑡, 2) 
e)r’(t)=(2𝑡, (sec 𝑡)2, 0), r’’(t)=(2,2(sec 𝑡)2𝑡𝑔 𝑡, 0) 
5 
a)3𝑡 + 𝑡𝑠𝑒𝑛𝑡 + 2𝑡2 
b)(𝑡𝑒−𝑡, 𝑒−𝑡𝑠𝑒𝑛𝑡, 2𝑒−𝑡) 
c)(−6 + 𝑡, −2𝑡 + 𝑠𝑒𝑛𝑡, 2 − 2𝑡2) 
d)(𝑡2𝑠𝑒𝑛𝑡 − 2𝑡)𝒊 + (6 − 𝑡3)𝒋 + (𝑡2 − 3𝑠𝑒𝑛𝑡)𝒌 
 
6 
a) 
F’(t)=(6𝑡, −𝑒−𝑡,
2𝑡
𝑡2+1
) 
F’’(t)=(6, 𝑒−𝑡,
−2𝑡2+2
(𝑡2+1)2
) 
b) 
F’(t)=(
2
3 √𝑡
3 ) 𝒊 + (−2𝑡𝑠𝑒𝑛𝑡
2)𝒋 + (3)𝒌 
F’’(t)=(
−2
9 √𝑡4
3 ) 𝒊 + (−2𝑠𝑒𝑛𝑡
2 − 4𝑡2𝑐𝑜𝑠𝑡2)𝒋 
c) 
F’(t)=(5𝑐𝑜𝑠5𝑡)𝒊 + (−4𝑠𝑒𝑛4𝑡)𝒋 + 2𝑒−2𝑡𝒌 
F’’(t)=(−25𝑠𝑒𝑛5𝑡)𝒊 + (−16𝑐𝑜𝑠4𝑡)𝒋−4𝑒−2𝑡𝒌 
 
7 
a)
1
2
𝒊 + (𝑒 − 1)𝒋 
b)
𝜋
2
𝒋 + 2𝒌 
c)3𝒊 + 2𝒋 + 𝒌 
 
8 
a)
1
2
𝒊 +
1
3
𝒋 +
1
4
𝒌 
b) 
10
3
𝒊 −
124
5
𝒋 −
4
3
𝒌 
c)
1
2
𝒊 +
1
2
𝒋 +
√2(𝜋−2)
4
𝒌 
 
9 
𝑡3
3
𝒊 + (𝑡4 + 1)𝒋 −
𝑡3
3
𝒌 
10 
(−𝑐𝑜𝑠𝑡 + 2)𝒊 + (−𝑠𝑒𝑛𝑡 + 1)𝒋 + (𝑡2 + 2)𝒌 
11) Calcule primeiro o produto escalar, depois derive 
1 − 4𝑡𝑐𝑜𝑠𝑡 + 11𝑡2𝑠𝑒𝑛𝑡 + 3𝑡³𝑐𝑜𝑠𝑡